MEDINAH, Ill. — Erasing some of their worst Ryder Cup memories, the Europeans wore the image of Seve Ballesteros on their sleeves and played their hearts out Sunday at Medinah to match the greatest comeback in history and head home with that precious gold trophy.

Europe got its payback for Brookline, when the Americans roared back from the same 10-6 deficit. This rally was even more remarkable, carried out before a raucous American crowd that began their chants of “USA!” some three hours before the first match got under way.

Jose Maria Olazabal squeezed his eyes and fought back tears when Kaymer holed a 6-foot par putt to beat Steve Stricker and give Europe the point it needed to keep the cup. This was the first Ryder Cup since Ballesteros, the soul of European golf in this event, died last May of a brain tumor. Olazabal wanted his team to wear navy blue, Seve’s favorite color, and added a clever touch — his iconic silhouette on the sleeves of their shirts.

“This one is for all of Europe,” Olazabal said. “Seve will always be present with this team. He was a big factor for this event for the European side, and last night when we were having that meeting, I think the boys understood that believing was the most important thing. And I think they did.”

Tiger Woods missed a 3½-foot par putt on the 18th hole, and then conceded a par to Francesco Molinari of about that length to halve their match. That extra half-point made it a clear-cut win for Europe, 14½-13½.

Woods and Stricker, the anchors in the lineup, didn’t win a single match at Medinah.

Ian Poulter was the first to embrace Olazabal, which was only fitting.

It was Poulter who gave Europe hope Saturday evening when he made five straight birdies to turn a loss into a win and swing momentum in Europe’s favor. Poulter was up to his fist-pumping, eye-bulging tricks again on the final day, winning the last two holes in his match against U.S. Open champion Webb Simpson.

And he had plenty of help. Europe’s top five players in the lineup all won, including Rory McIlroy, who was lucky to be playing. McIlroy thought his match was at 12:25 p.m. — it was listed in Eastern time, not Central — and needed a police escort to get to the course with 10 minutes to spare. Then, he came up with key birdies to hand Keegan Bradley his first loss of the week.

The biggest match might have belonged to Justin Rose. He was on the verge of losing to Phil Mickelson when Rose holed a 12-foot par putt to halve the 16th, made a 35-foot birdie putt from the back of the 17th green to win the hole, and then closed out Mickelson with a 12-foot birdie on the last hole. Six of the 12 matches went to the 18th hole on Sunday. The Americans won only one of them.

The Americans also rallied from a four-point deficit to win in 1999 at Brookline. This was different, though. The Americans won big in those early matches. At Medinah, so many of them could have gone either way.

It was so close, so tense, that either side could have won the Ryder Cup down to the very end.

Stricker made an 8-foot par putt on the 18th, and Kaymer faced a par putt from 6 feet to win the match. If he missed, the Americans would get a half-point, and Woods was leading 1-up over Molinari and in the middle of the 18th fairway.

Kaymer, a former No. 1 and major champion who has struggled all year, poured it in the middle and the celebration was on.

He could barely speak at this point, not so much from pure emotion but having to scream over the crowd behind him. Players were hugging and crying, and the small European contingent that had been drowned out all week was serenading themselves with what has become the theme song of the Ryder Cup.

“Ole, ole, ole, ole,” they sang merrily, even as the teams prepared for the closing ceremony.

Europe now has won seven of the last nine Ryder Cups, and even more remarkable about this comeback is that they did it on the road.

Davis Love III became the first U.S. captain to sit every player at least once before Sunday, wanting them to be fresh for the decisive day. Instead, the Americans faltered at the end — especially Jim Furyk and Stricker, two of his captain’s picks.

“The plan worked the first two days,” he said. “It just didn’t work today.”

The only U.S. points came from Dustin Johnson, who went 3-0 in this Ryder Cup, Zach Johnson and unheralded Jason Dufner.

“We’re all kind of stunned,” Love said. “We know what it feels like now from the ‘99 Ryder Cup. It’s a little bit shocking. We were playing so well, we figured it didn’t matter how we sent them out there. We got a couple of matches flipped there in the middle that cost us.”

Love thought all along the Ryder Cup would be decided in the ninth match by Dufner. It was most appropriate that Europe won the cup thanks to Kaymer.

Kaymer gave German golf some redemption from Kiawah Island in 1991, when countryman Bernhard Langer missed a par putt from about the same length that allowed the Americans to win.

“It’s a feeling I never had before,” Kaymer said. “On Friday, I sat down with Bernhard and talked a little bit about the Ryder Cup because my attitude was not the right one. But now I know how important the Ryder Cup is.”